A converse theorem and the Saito-Kurokawa lift. (English) Zbl 0849.11039

Duke, W.; Imamoḡlu, Özlem

Int. Math. Res. Not. 1996, No. 7, 347-355 (1996).

In the paper under review the authors give a new proof of the Saito-Kurokawa lifting. More precisely they construct the lifting from Kohnen’s \(+\)-space to the Maaß Spezialschar avoiding Jacobi forms. The new approach is to use K. Imai’s converse theorem [Am. J. Math. 102, 903-936 (1980; Zbl 0447.10028)], i.e. a generalization of Hecke’s correspondence between Siegel cusp forms of degree 2 and Koecher-Maaß series with functional equations. This non-trivial reduction requires the identification of a sum over Heegner points of a Maaß form of weight 0 as Fourier coefficients of a form of weight 1/2.

Reviewer: A.Krieg (Aachen)

MSC:

11F46	Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F37	Forms of half-integer weight; nonholomorphic modular forms

Keywords:

Siegel modular forms; Maaß wave form; Imai’s converse theorem; Saito-Kurokawa lifting; Siegel cusp forms; Koecher-Maaß series